Séminaire Probabilités et Statistiques
Sparse expanders have negative curvature
02
déc. 2021
déc. 2021
Intervenant : | Justin Salez |
Institution : | Ceremade (Université Paris Dauphine) |
Heure : | 14h00 - 15h00 |
Lieu : | 3L15 |
We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). To establish this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible “at infinity”. We then transfer this result to finite graphs via local weak convergence. The same approach applies to the Bakry-Émery curvature condition CD(0, ∞), thereby settling a recent conjecture of Cushing, Liu and Peyerimhoff (2019).